2.2 problem 9

Internal problem ID [823]
Internal file name [OUTPUT/823_Sunday_June_05_2022_01_50_35_AM_63213515/index.tex]

Book: Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section: Chapter 4.2, Higher order linear differential equations. Constant coefficients. page 180
Problem number: 9.
ODE order: 3.
ODE degree: 1.

The type(s) of ODE detected by this program : "higher_order_linear_constant_coefficients_ODE"

Maple gives the following as the ode type

[[_3rd_order, _missing_x]]

\[ \boxed {y^{\prime \prime \prime }-3 y^{\prime \prime }+3 y^{\prime }+y=0} \] The characteristic equation is \[ \lambda ^{3}-3 \lambda ^{2}+3 \lambda +1 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -2^{\frac {1}{3}}+1\\ \lambda _2 &= \frac {2^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1\\ \lambda _3 &= \frac {2^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1 \end {align*}

Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{\left (\frac {2^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1\right ) x} c_{1} +{\mathrm e}^{\left (\frac {2^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1\right ) x} c_{2} +{\mathrm e}^{\left (-2^{\frac {1}{3}}+1\right ) x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{\left (\frac {2^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1\right ) x}\\ y_2 &= {\mathrm e}^{\left (\frac {2^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1\right ) x}\\ y_3 &= {\mathrm e}^{\left (-2^{\frac {1}{3}}+1\right ) x} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{\left (\frac {2^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1\right ) x} c_{1} +{\mathrm e}^{\left (\frac {2^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1\right ) x} c_{2} +{\mathrm e}^{\left (-2^{\frac {1}{3}}+1\right ) x} c_{3} \\ \end{align*}

Verification of solutions

\[ y = {\mathrm e}^{\left (\frac {2^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1\right ) x} c_{1} +{\mathrm e}^{\left (\frac {2^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1\right ) x} c_{2} +{\mathrm e}^{\left (-2^{\frac {1}{3}}+1\right ) x} c_{3} \] Verified OK.

2.2.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }-3 y^{\prime \prime }+3 y^{\prime }+y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & y^{\prime \prime \prime } \\ \square & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{1}\left (x \right ) \\ {} & {} & y_{1}\left (x \right )=y \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{2}\left (x \right ) \\ {} & {} & y_{2}\left (x \right )=y^{\prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{3}\left (x \right ) \\ {} & {} & y_{3}\left (x \right )=y^{\prime \prime } \\ {} & \circ & \textrm {Isolate for}\hspace {3pt} y_{3}^{\prime }\left (x \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & y_{3}^{\prime }\left (x \right )=3 y_{3}\left (x \right )-3 y_{2}\left (x \right )-y_{1}\left (x \right ) \\ & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & {} & \left [y_{2}\left (x \right )=y_{1}^{\prime }\left (x \right ), y_{3}\left (x \right )=y_{2}^{\prime }\left (x \right ), y_{3}^{\prime }\left (x \right )=3 y_{3}\left (x \right )-3 y_{2}\left (x \right )-y_{1}\left (x \right )\right ] \\ \bullet & {} & \textrm {Define vector}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}\left (x \right )=\left [\begin {array}{c} y_{1}\left (x \right ) \\ y_{2}\left (x \right ) \\ y_{3}\left (x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (x \right )=\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -1 & -3 & 3 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (x \right ) \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -1 & -3 & 3 \end {array}\right ] \\ \bullet & {} & \textrm {Rewrite the system as}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (x \right )=A \cdot {\moverset {\rightarrow }{y}}\left (x \right ) \\ \bullet & {} & \textrm {To solve the system, find the eigenvalues and eigenvectors of}\hspace {3pt} A \\ \bullet & {} & \textrm {Eigenpairs of}\hspace {3pt} A \\ {} & {} & \left [\left [-2^{\frac {1}{3}}+1, \left [\begin {array}{c} \frac {1}{\left (-2^{\frac {1}{3}}+1\right )^{2}} \\ \frac {1}{-2^{\frac {1}{3}}+1} \\ 1 \end {array}\right ]\right ], \left [\frac {2^{\frac {1}{3}}}{2}-\frac {\mathrm {I} \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1, \left [\begin {array}{c} \frac {1}{\left (\frac {2^{\frac {1}{3}}}{2}-\frac {\mathrm {I} \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1\right )^{2}} \\ \frac {1}{\frac {2^{\frac {1}{3}}}{2}-\frac {\mathrm {I} \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1} \\ 1 \end {array}\right ]\right ], \left [\frac {2^{\frac {1}{3}}}{2}+\frac {\mathrm {I} \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1, \left [\begin {array}{c} \frac {1}{\left (\frac {2^{\frac {1}{3}}}{2}+\frac {\mathrm {I} \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1\right )^{2}} \\ \frac {1}{\frac {2^{\frac {1}{3}}}{2}+\frac {\mathrm {I} \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-2^{\frac {1}{3}}+1, \left [\begin {array}{c} \frac {1}{\left (-2^{\frac {1}{3}}+1\right )^{2}} \\ \frac {1}{-2^{\frac {1}{3}}+1} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}={\mathrm e}^{\left (-2^{\frac {1}{3}}+1\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (-2^{\frac {1}{3}}+1\right )^{2}} \\ \frac {1}{-2^{\frac {1}{3}}+1} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [\frac {2^{\frac {1}{3}}}{2}-\frac {\mathrm {I} \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1, \left [\begin {array}{c} \frac {1}{\left (\frac {2^{\frac {1}{3}}}{2}-\frac {\mathrm {I} \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1\right )^{2}} \\ \frac {1}{\frac {2^{\frac {1}{3}}}{2}-\frac {\mathrm {I} \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (\frac {2^{\frac {1}{3}}}{2}-\frac {\mathrm {I} \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (\frac {2^{\frac {1}{3}}}{2}-\frac {\mathrm {I} \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1\right )^{2}} \\ \frac {1}{\frac {2^{\frac {1}{3}}}{2}-\frac {\mathrm {I} \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Use Euler identity to write solution in terms of}\hspace {3pt} \sin \hspace {3pt}\textrm {and}\hspace {3pt} \cos \\ {} & {} & {\mathrm e}^{\left (\frac {2^{\frac {1}{3}}}{2}+1\right ) x}\cdot \left (\cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (\frac {2^{\frac {1}{3}}}{2}-\frac {\mathrm {I} \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1\right )^{2}} \\ \frac {1}{\frac {2^{\frac {1}{3}}}{2}-\frac {\mathrm {I} \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (\frac {2^{\frac {1}{3}}}{2}+1\right ) x}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )}{\left (\frac {2^{\frac {1}{3}}}{2}-\frac {\mathrm {I} \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1\right )^{2}} \\ \frac {\cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )}{\frac {2^{\frac {1}{3}}}{2}-\frac {\mathrm {I} \sqrt {3}\, 2^{\frac {1}{3}}}{2}+1} \\ \cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{y}}_{2}\left (x \right )={\mathrm e}^{\left (\frac {2^{\frac {1}{3}}}{2}+1\right ) x}\cdot \left [\begin {array}{c} \frac {\sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right ) 2^{\frac {2}{3}} \sqrt {3}-2^{\frac {2}{3}} \cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )+2 \sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right ) 2^{\frac {1}{3}} \sqrt {3}+2 \,2^{\frac {1}{3}} \cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )+2 \cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )}{2 \left (2^{\frac {2}{3}}+2^{\frac {1}{3}}+1\right )^{2}} \\ \frac {\sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right ) 2^{\frac {1}{3}} \sqrt {3}+2^{\frac {1}{3}} \cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )+2 \cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )}{2 \left (2^{\frac {2}{3}}+2^{\frac {1}{3}}+1\right )} \\ \cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{3}\left (x \right )={\mathrm e}^{\left (\frac {2^{\frac {1}{3}}}{2}+1\right ) x}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right ) 2^{\frac {2}{3}} \sqrt {3}+2^{\frac {2}{3}} \sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )+2 \cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right ) 2^{\frac {1}{3}} \sqrt {3}-2 \,2^{\frac {1}{3}} \sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )-2 \sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )}{2 \left (2^{\frac {2}{3}}+2^{\frac {1}{3}}+1\right )^{2}} \\ \frac {\cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right ) 2^{\frac {1}{3}} \sqrt {3}-2^{\frac {1}{3}} \sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )-2 \sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )}{2 \left (2^{\frac {2}{3}}+2^{\frac {1}{3}}+1\right )} \\ -\sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right ) \end {array}\right ]\right ] \\ \bullet & {} & \textrm {General solution to the system of ODEs}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}=c_{1} {\moverset {\rightarrow }{y}}_{1}+c_{2} {\moverset {\rightarrow }{y}}_{2}\left (x \right )+c_{3} {\moverset {\rightarrow }{y}}_{3}\left (x \right ) \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}=c_{1} {\mathrm e}^{\left (-2^{\frac {1}{3}}+1\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (-2^{\frac {1}{3}}+1\right )^{2}} \\ \frac {1}{-2^{\frac {1}{3}}+1} \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{\left (\frac {2^{\frac {1}{3}}}{2}+1\right ) x}\cdot \left [\begin {array}{c} \frac {\sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right ) 2^{\frac {2}{3}} \sqrt {3}-2^{\frac {2}{3}} \cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )+2 \sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right ) 2^{\frac {1}{3}} \sqrt {3}+2 \,2^{\frac {1}{3}} \cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )+2 \cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )}{2 \left (2^{\frac {2}{3}}+2^{\frac {1}{3}}+1\right )^{2}} \\ \frac {\sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right ) 2^{\frac {1}{3}} \sqrt {3}+2^{\frac {1}{3}} \cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )+2 \cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )}{2 \left (2^{\frac {2}{3}}+2^{\frac {1}{3}}+1\right )} \\ \cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right ) \end {array}\right ]+c_{3} {\mathrm e}^{\left (\frac {2^{\frac {1}{3}}}{2}+1\right ) x}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right ) 2^{\frac {2}{3}} \sqrt {3}+2^{\frac {2}{3}} \sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )+2 \cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right ) 2^{\frac {1}{3}} \sqrt {3}-2 \,2^{\frac {1}{3}} \sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )-2 \sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )}{2 \left (2^{\frac {2}{3}}+2^{\frac {1}{3}}+1\right )^{2}} \\ \frac {\cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right ) 2^{\frac {1}{3}} \sqrt {3}-2^{\frac {1}{3}} \sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )-2 \sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )}{2 \left (2^{\frac {2}{3}}+2^{\frac {1}{3}}+1\right )} \\ -\sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right ) \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=-2 \left (\left (-c_{3} \sqrt {3}+c_{2} \right ) 2^{\frac {1}{3}}+\frac {3 \left (c_{3} \sqrt {3}+c_{2} \right ) 2^{\frac {2}{3}}}{4}-\frac {5 c_{2}}{2}\right ) {\mathrm e}^{\frac {\left (2^{\frac {1}{3}}+2\right ) x}{2}} \cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )+2 \left (\left (\sqrt {3}\, c_{2} +c_{3} \right ) 2^{\frac {1}{3}}+\frac {3 \left (-\sqrt {3}\, c_{2} +c_{3} \right ) 2^{\frac {2}{3}}}{4}-\frac {5 c_{3}}{2}\right ) {\mathrm e}^{\frac {\left (2^{\frac {1}{3}}+2\right ) x}{2}} \sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )+4 c_{1} \left (2^{\frac {1}{3}}+\frac {3 \,2^{\frac {2}{3}}}{4}+\frac {5}{4}\right ) {\mathrm e}^{-\left (2^{\frac {1}{3}}-1\right ) x} \end {array} \]

Maple trace

`Methods for third order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
checking if the LODE has constant coefficients 
<- constant coefficients successful`
 

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 58

dsolve(diff(y(x),x$3)-3*diff(y(x),x$2)+3*diff(y(x),x)+y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\left (2^{\frac {1}{3}}-1\right ) x}+c_{2} {\mathrm e}^{\frac {\left (2^{\frac {1}{3}}+2\right ) x}{2}} \sin \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right )+c_{3} {\mathrm e}^{\frac {\left (2^{\frac {1}{3}}+2\right ) x}{2}} \cos \left (\frac {2^{\frac {1}{3}} \sqrt {3}\, x}{2}\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 87

DSolve[y'''[x]-3*y''[x]+3*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+1\&,1\right ]\right )+c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+1\&,2\right ]\right )+c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+1\&,3\right ]\right ) \]